7
votes
1answer
267 views

Quasi-isomorphisms in exact categories

I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...
3
votes
3answers
266 views

When an exact embedding of abelian categories induces a full embedding of their derived categories?

Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also? I would be interested in any necessary or sufficient ...
7
votes
1answer
242 views

Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...
2
votes
1answer
124 views

Can I bound the degree of a contracting homotopy in an exact filtered complex?

Suppose that I have given you a bigraded vector space $V = \bigoplus_{i,j} V_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. ...
8
votes
0answers
203 views

(Reduced) cyclic homology of a free product of unital algebras

Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...
43
votes
1answer
3k views

Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories. Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...
11
votes
4answers
707 views

Injective dimension of graded-injective modules.

In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra": Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with ...
6
votes
2answers
393 views

Somewhat general question that includes: “Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?”

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree? Example: Given a dg ...
3
votes
1answer
406 views

Local Cohomology and Maximal-Cohen-Macaulay modules

Checking a recent article [this one, specifically section 3.1] I found the following claim (I'm paraphrasing, of course): Let $A$ be a graded connected noetherian algebra (not necessarily ...
22
votes
2answers
1k views

Where do all these projection formulas come from?

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples. Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, ...
3
votes
3answers
1k views

Homology or cohomology?

How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor ...
56
votes
4answers
4k views

Algorithm or theory of diagram chasing

One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ...
38
votes
5answers
5k views

Why are spectral sequences so ubiquitous?

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...
6
votes
1answer
751 views

Sources for exact triangles in triangulated categories.

The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides ...